Related papers: Bases in the solution space of the Mellin system
Let $Z\subset{\bf P}^{n-1}$ be a hypersurface such that the associated reduced hypersurface $Z_{\rm red}$ has only weighted homogeneous isolated singularities. In the case $Z$ is a reduced curve or $Z_{\rm red}$ has only homogeneous…
The algebraic matrix hierarchy approach based on affine Lie $sl (n)$ algebras leads to a variety of 1+1 soliton equations. By varying the rank of the underlying $sl (n)$ algebra as well as its gradation in the affine setting, one…
For a complex polynomial or analytic function f, one has been studying intensively its so-called local zeta functions or complex powers; these are integrals of |f|^{2s}w considered as functions in s, where the w are differential forms with…
The existence of sufficiently many finite order meromorphic solutions of a differential equation, or difference equation, or differential-difference equation, appears to be a good indicator of integrability. In this paper, we investigate…
The classical construction of the Weil representation, with complex coefficients, has long been expected to work for more general coefficient rings. This paper exhibits the minimal ring $\mathcal{A}$ for which this is possible, the integral…
Here the polynomial interpolation approach is used to introduce the main results on multivariate normal algebraic systems. Next we bring a construction which shows that any standard algebraic system, with finite set of solutions, can be…
We prove a weak fundamental principle for $\lambda$-homogeneous solutions of homogeneous constant-coefficient systems on open pointed convex cones. Starting with the solution family $S_{\mathcal B}$ arising in the Ehrenpreis--Palamodov…
Based on the recent progress in the irregular Riemann-Hilbert correspondence, we study the monodromies at infinity of the holomorphic solutions of Fourier transforms of holonomic D-modules in some situations. Formulas for their eigenvalues…
In this work, we revisit the study by M. E. Schonbek [11] concerning the problem of existence of global entropic weak solutions for the classical Boussinesq system, as well as the study of the regularity of these solutions by C. J. Amick…
We reveal a complexity chasm, separating the trinomial and tetranomial cases, for solving univariate sparse polynomial equations over certain local fields. First, for any fixed field $K\in\{\mathbb{Q}_2,\mathbb{Q}_3,\mathbb{Q}_5,\ldots\}$,…
A tautological system, introduced in \cite{LSY}\cite{LY}, arises as a regular holonomic system of partial differential equations that govern the period integrals of a family of complete intersections in a complex manifold $X$, equipped with…
We prove that, given a path of Beltrami differentials on $\mathbb C$ that live in and vary holomorphically in the Sobolev space $W^{l,\infty}_{loc}(\Omega)$ of an open subset $\Omega\subset \mathbb C$, the canonical solutions to the…
In this paper we consider monic polynomials such that their coefficients coincide with their zeros. These polynomials were first introduced by S. Ulam. We combine methods of algebraic geometry and dynamical systems to prove several results.…
Linear differential equations with polynomial coefficients over a field $K$ of positive characteristic $p$ with local exponents in the prime field have a basis of solutions in the differential extension $\mathcal{R}_p=K(z_1, z_2,…
We obtain quantitative estimates of local flatness of zero sets of harmonic polynomials. There are two alternatives: at every point either the zero set stays uniformly far away from a hyperplane in the Hausdorff distance at all scales or…
A multidimensional generalization of Melvin's solution for an arbitrary simple Lie algebra $\cal G$ is considered. The gravitational model in $D$ dimensions, $D \geq 4$, contains $n$ 2-forms and $l \geq n$ scalar fields, where $n$ is the…
We complete the complexity classification by degree of minimizing a polynomial over the integer points in a polyhedron in $\mathbb{R}^2$. Previous work shows that optimizing a quadratic polynomial over the integer points in a polyhedral…
We argue that the Mellin-Barnes representations of Feynman diagrams can be used for obtaining linear systems of homogeneous differential equations for the original Feynman diagrams with arbitrary powers of propagators without recourse to…
This paper deals with the solution of large classes of systems of nonlinear partial differential equations (PDEs) in spaces of generalized functions that are constructed as the completion of uniform convergence spaces. The existence result…
We consider the KZ differential equations over $\mathbb C$ in the case, when its multidimensional hypergeometric solutions are one-dimensional integrals. We also consider the same differential equations over a finite field $\mathbb F_p$. We…